Near-squares in binary recurrence sequences

Abstract

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a\ge 3$ by $u_0(a)=0$, $u_1(a)=1$ and $u_{n+2}(a)=a{u}_{n+1}(a)-u_n(a)$ for $n\ge 0$. We show that for a given $a\ge 3$, there is at most one $n\ge 5$ such that $u_n(a)$ is a near-square. With the exceptions of $u_6(3)=12^2$ and $u_7(6)=239\cdot 13^2$, any such $u_n(a)$ can be a near-square only if $a\equiv 2(mod4)$, $n\equiv 3(mod4)$ is prime and $n\ge 19$.

This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers $a$ and $b=-b_1^2$ by $u_0(a,b)=0$, $u_1(a,b)=1$ and $u_{n+2}(a,b)=a{u}_{n+1}(a,b)+b{u}_n(a,b)$ for $n\ge 0$. This arises from a novel Aurifeuillean-type factorization of $u_{2n+1}(a,b)$ we have found.